Question

example 4

john graphed the function \\(f(x) = 2x^3 + 5x^2 - 28x - 15\\) and his friend jose graphed the function \\(g(x) = (x + 4)(x + 1)(x - 2)(x - 4)\\). whose function has more zeros?

solution

zeros of a function are the values of \\(x\\) which make the function equal to zero.

johns function, \\(f(x) = 2x^3 + 5x^2 - 28x - 15\\)

value: 1

find the degree of johns function.

a. 1
b. 2
c. 3
d. 4

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Degree of Polynomial",
"Polynomial Functions"
],
"current_concepts": [
"Degree of Polynomial",
"Polynomial Functions"
]
}
</pre_analysis>

<reasoning>

Identify the polynomial function

The function given for John is:
\[f(x) = 2x^3 + 5x^2 - 28x - 15\]

Determine the highest exponent of the variable

The terms of the polynomial are \(2x^3\), \(5x^2\), \(-28x\), and \(-15\). The exponents of \(x\) in these terms are \(3\), \(2\), \(1\), and \(0\) respectively. The highest exponent is \(3\).

State the degree of the polynomial

The degree of a polynomial is the highest exponent of its variable. Therefore, the degree of John's function is \(3\).
</reasoning>

<answer>
<mcq-option>a. 1</mcq-option>
<mcq-option>b. 2</mcq-option>
<mcq-correct>c. 3</mcq-correct>
<mcq-option>d. 4</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Degree of Polynomial"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Degree of Polynomial",
"Polynomial Functions"
],
"current_concepts": [
"Degree of Polynomial",
"Polynomial Functions"
]
}
</pre_analysis>

<reasoning>

Identify the polynomial function

The function given for John is:
\[f(x) = 2x^3 + 5x^2 - 28x - 15\]

Determine the highest exponent of the variable

The terms of the polynomial are \(2x^3\), \(5x^2\), \(-28x\), and \(-15\). The exponents of \(x\) in these terms are \(3\), \(2\), \(1\), and \(0\) respectively. The highest exponent is \(3\).

State the degree of the polynomial

The degree of a polynomial is the highest exponent of its variable. Therefore, the degree of John's function is \(3\).
</reasoning>

<answer>
<mcq-option>a. 1</mcq-option>
<mcq-option>b. 2</mcq-option>
<mcq-correct>c. 3</mcq-correct>
<mcq-option>d. 4</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Degree of Polynomial"
]
}
</post_analysis>